Non-Well-Founded Sets

NON-WELL-FOUNDED SETS

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CSLI Lecture Notes Number 14

NON-WELL-FOUNDED SETS

Peter Aczel Foreword by Jon Barwise

CENTER FOR THE STUDY OF LANGUAGE

AND INFORMATION

CSLI was founded early in 1983 by researchers from Stanford University, SRI International, and Xerox PARC to further research and development of integrated theories of language, information, and computation. CSLI headquarters and the publication offices are located at the Stanford site .

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Copyright

© 1988

Center for the Study of Language and Information Leland St anford Junior University Printed in the United States 94 93 92 91 90 89 88

5 4

3

2 1

Library of Con g ress Cataloging-in-Publication Data

Aczel, Peter, 1941Non-well-founded sets. (CSLI lecture notes; no. 14 ) Bibliography: p. Includes index. 1. Axiomatic set theory. 1. Title. II. Series.

Q A248.A28

1988 511.3'22

ISBN 0-937073-21-0 ISBN 0-937073-22-9 ( pbk.)

87-17857

To my mother and father, Suzy and George Aczel

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Let E be a set, E' one of its elements, E" any element of E',

so on. I call a E", etc.

descent the

and

sequence of steps from E to E', E' to

... I say that a set is ordinary when it only gives rise to finite descents; I say t hat it is extraordinary when among its .

descents there are some which are infinite.

-Mirimanoff (1917) Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles

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Contents

Foreword Xl Preface Xlll Introduction xvii

1 2 3

4 5

6 7 8

A B

Part One The Anti-Foundation Axiom Introducing the Axiom 3 The Axiom in More Detail 19 A Model of the Axiom 33

1

Part Two Variants of the Anti-Foundation Axiom Variants Using a Regular Bisimulation 41 Another Variant 57

39

Part Three On Using the Anti-Foundation Axiom 71 Fixed Points of Set Continuous Operators 73 The Special Final Coalgebra Theorem 8 1 An Application to Communicating Systems 91 Appendices 101 Notes Towards a History 103 Background Set Theory 113

References 123 Index of Definitions 1 29 Index of Named Axioms and Results

131

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Foreword

To my way of thinking, mathemtical logic is a branch of applied mathematics. It applies mathematics to model and study various sorts of symbolic systems: axioms, proofs, programs, computers, or people talking and reasoning together. This is the only view of mathematical logic which does justice to the logician's intuition that logic really is a field, not j ust the union of several unrelated fields. One expects that logic, as a branch of applied mathematics, will not only use existing tools from mathematics, but also that it will lead to the creation of new mathematical tools, tools that arise out of the need to model some real world phenomena not ad­ equately modeled by previously known mathematical structures. Turing's analysis of the notion of algorithm by means of Turing machines is an obvious example. In this way, by reaching out and st udy ing new pheonemona, applied mathematics in general, and mathematical logic in particular, enriches mathematics, not only with new theorems, but also with new mathematical structures, structures for the mathematician to study and for others to apply in new domains. The th eory of circular and otherwise extra-ordinary sets pre­ sented in this book is an excellent example of this synergistic process. Aczel's work was motivated by work of Robin Milner in computer science modeling concurrent processes. The fact that these processes are inherently circular makes them awkward to model in traditonal set theory, since most straightforward ideas run afoul of the axiom of foundation. As a result , Milner's own treatment was highly syntactic. Aczel's original aim was to find a version of set theory where these circular phenomena could be modeled in a straightforward way, using standard techniques from set theory. This forced him to develop an alternative conception Xl

xii

Foreword

of set, the conception t hat lies at the heart of this book. Aczel returns to his starting point in the final chapter of this book. Before learning of Aczel's work, I had run u p against similar difficulties in my work in situation theory and situation seman­ tics. It seemed that in order to understand common knowledge (a crucial feature of communication) , circular propositions, vari­ ous aspects of perceptual knowledge and self-awareness, we had to admit that there are situations that are not wellfounded under the "constituent of" relation. This meant that the most natural route to modeling situations was blocked by the axiom of founda­ tion. As a result , we either had to give up the tools of set theory which are so well loved in mathematical logic, or we had to enrich the conception of set , finding one that admits of circular sets, at least. I wrestled with this dilemma for well over a year before I argued for the latter move in (Barwise 1986). It was at j ust this point that Aczel visited CSLI and gave the seminar which formed the basis of this book. Since then, I have found several applications of Aczel's set theory, far removed from the problems in computer science that originally motivated Aczel . Others have gone on to do interesting work of a strictly mathematical nature exploring this expanded universe of sets. I feel quite certain that there is still a lot to be done with this universe of sets, on both fronts, that there are mathematical prob­ lems to be solved, and further applications to be found. However, there is a serious linguistic obstacle to this work, arising out of the dominance of the cumulative conception of set. Just as there used to be complaints about referring to complex numbers as numbers, so there are objections to referring to non-well-founded sets as sets. While there is clear historical j ustification for this usage, the objection persists and distracts from the interest and importance of the subject . However, I am convinced that readers who approach this book unencumbered by this linguistic prob­ lem will find themselves amply rewarded for their effort . The AFA theory of non-well-founded sets is a beautiful one, full of p o­ tential for mathematics and its applications to symbolic systems. I am delighted to have played a small role, as Director of CSLI during Aczel's stay, in helping to bring this book into existence.

JON BARWISE

Preface

This work started out as lecture notes for a graduate course called "Sets and Processes" given in the Mathematics Department at Stanford University in the period January-March 1 985. In fact some of the material was handed out week by week as the course progressed and was then tidied up to form the first draft of the book. Jon Barwise suggested that this should appear in the CSLI lecture notes series, and I eagerly and gratefully accepted his suggestion. I think that the idea in both our minds at the time was that a few more weeks of work on the notes would put them in suitable shape for publication. This idea turned out to be mistaken and several years have passed. I hope that the book is somewhat better than it would have been had I finished it according to plan. In many respects the book follows the structure of the origi­ nal draft. In particular there are exercises scattered through the text which were put in for a variety of reasons. Some of them are essential for what follows, but only involve routine arguments. Others make points that are less essential. Parts one and two con­ tain a revised version of the original draft. Part three contains work related to material that was given in the original lectures, but in much more detail. In addition I have added an appendix, Notes Towards a History, in which I attempt to present the his­ torical information I have gathered in working on the book. A surprising range of people have written about non-wen-founded sets, often in ignorance of each other. I have tracked down some of this work. No doubt there are yet more discoveries to be made here. In an attempt to give the book a wider readership I have in­ cluded an appendix, Background Set Theory, in which the reader can find out what are the set theoretical notation and ideas that Xlll

XIV

Preface

are needed to read the book. The reader is recommended to start out directly with chapter one which requires only a limited fa­ miliarity with set theory. Later chapters require more familiarity and the reader should consult the appendix as necessary. This book would never have appeared without the initial sug­ gestion and continual encouragement of Jon Barwise. He has displayed great patience and also imagination in finding new rea­ sons why the book ought to be finished. More importantly he has discovered an area of application for non-well-founded sets, Situation Theory, which promises to make these sets of wider in­ terest than I could have envisaged when I first became interested in them. That application is barely treated in this book. For that the reader should refer to some of his publications. In particular the book, The Liar, written by John Etchemendy and Jon Bar­ wise, can usefully be read in conj unction with this book. That book gives an elegant intuitive treatment of the anti-foundation axiom and makes an interesting application of it to the philo­ sophical problem of the liar paradox. As usual there are a variety of people and institutions who had a hand, sometimes unforseen, in the completion of this book. The excellent physical and intellectual environment of Stanford University and eSLI was an important stimulus for me. I am grateful to the System Development Foundation who funded my visit to eSLI during the period October 84-April 85. It was there that I first encountered the marvelous 'lEX typesetting system. I learnt to use IffiTEX , the package of 'lEX macros designed by Leslie Lamport, by using it to write the original drafts of this book, and have continued to use it since with great enthusiasm. Donald Knuth deserves the gratitude of many people like myself for his creation of the 'lEX system. After visiting Stanford I took up a 3 month SERe research position at Edinburgh University during the summer of 1985. I am grateful to Gordon Plotkin for organising this, and also to him and Robin Milner for discussions with them on concurrent processes while I was there. I came to learn that the notion of a concurrent process was a good deal more complex and subtle than I had thought when I first started to think about the notion and its relationship to non-well-founded sets. Robin Milner's work on sees was the direct cause for my original interest in non-well-founded sets.

Preface

xv

Some of the writing of this book took place while I was on leave from the Mathematics Department at Manchester Univer­ sity, with a research position at the Computer Science Depart­ ment of Manchester University, during parts of 1986 and 1987. I am grateful to ICL for the funding of this arrangment . I would like to thank Emma Pease at CSLI, for her work in t ranslat i n g the IffiTEX files of this book into the appropriate 1EX files used in the CSLI Lecture Notes Series, and to Judy Boyd at Manchester University, for her help with some of the Iffi.TEX typing. Dikran Karagueuzian managed the production of this book and I am grateful for his advice and patient assistance at the various stages of the book's progress. I thank my wife Helen for her continual encouragement. She shared with me the ups and downs involved in the completion of this book. Finally there is lovely Rosalind. She cannot really be blamed for the further final delay that marked her first six months of life. Manchester 24 December 1981

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Introduction

A

non-well-founded set is an extraordinary set in the sense of Mirimanoff. * Such a set has an infinite descending membership sequence; i.e. an infinite sequence of sets, consisting of an ele­ ment of the set, an element of that element, an element of that element of that element and so on ad infinitum. What is ex­ traordinary about such a set is that it would seem that it could never get formed; for in order to form the set we would first have to form its elements, and to form those elements we would have to have previously formed their elements and so on leading to an infinite regress. Of course this anthropomorphic manner of speaking about the formation of sets is only that; a manner of speaking. We humans do not actually physically form sets out of their elements, as sets are abstract obj ects . Nevertheless the sets that have been needed to represent the standard abstract objects of modern mathematics have , in fact, been the ordinary well-founded ones. This observation has been institutionalised in the standard axiom system ZFC of axiomatic set theory, which includes among its axioms the foundation axiom FA. This axiom simply expresses that all sets are well-founded. If non-well-founded sets are not needed for the development of mathematics then it may well seem natural to leave them out of consideration when formulating an axiomatic basis for math­ ematics. Sometimes a stronger view is expressed. According to that view there is only one sensible coherent notion of set. That is the iterative conception in which sets are arranged in levels, with the elements of a set placed at lower levels than the set itself. For the iterative conce ption only well-founded sets exist and FA and the other axioms of ZFC are true when interpreted in the itera­ tive universe of pure sets. There has been yet one more reason * See the epigraph. XVII

XVlll

Introduction

why FA has been routinely included among the axioms of ax­ iomatic set theory. This is the fact that the cumulative hierarchy of the iterative universe has an enticingly elegant mathematical structure. This structure was already revealed by Mirimanoff and over the years it has been powerfully exploited by set theorists in a great variety of model constructions. There is a natural re­ luctance to forgo the pleasure of working within this structure. Certainly I myself must admit to having been rather seduced by it. But there have been doubts about the coherence of the itera­ tive conception. Part of its appeal has been the essentially naive but very intuitive image of a set being physically formed out of its elements. This image is translated to an abstract realm and given some plausibility by a sometimes subconcious suggestion of constructivity. The suggestion is to take the abstract realm to be a realm of mental constructions . In fact such a sugges­ tion cannot easily be sustained ( not by me anyway ) and one is forced to a Platonistic conception in which sets are taken to have a non-physical existence independent of us. The purpose of Part One of this book is to investigate the ax­ iom system obtained by replacing FA in ZFC by an axiom that I have chosen to call the Anti-Foundation Axiom, abbreviated A FA . This axiom expresses, in a particular way, that every pos­ sible non-well-founded set exists. The resulting axiom system is ZFC- + A FA , where ZFC- is ZFC without FA . There are other variants to A FA which also can be viewed as expressing that every possible non-well-founded set exists. I call these axioms collectively anti-foundation axioms, reserving A FA for the anti-foundation axiom. Some of these variants are discussed in Part Two. The reason for the existence of more than one anti-foundation axiom is the fact that there is more than one criterion for equality between sets when non-well-founded sets are allowed. One approach is to keep to the extensionality criterion of ZFC as the sole one, even for possibly non-well-founded sets; so sets are equal if they have the same elements and nothing further is to be stipulated concerning set equality. This is the approach that was developed in a series of papers in the 1 960s and early 1970s by Maurice Boffa and is here presented in chapter 5, where we call the resulting anti-foundation axiom BA FA . The other variants of AFA do use a strengthening of extensionality as a cri­ terion for set equality. It turns out that these other approaches

Introduction

XIX

can be treated in a uniform manner and this leads to the formu­ lation of a!l axiom A FA� relative to the definition of a suitable to express the criterion of set equality. This is pre­ relation sented in Chapter 4. The suitable relations are called regular bisimulation relations and range between two possible extremes. One extreme is the maximal bisimulation relation on the uni­ verse of sets. This relation gives the most generous criterion for set equality which roughly states that sets are equal whenever possible, keeping in mind that if two sets are equal then any el­ ement of one set must be equal to an element of the other set . It is this relation that gives rise to the axiom AFA . There is the other extreme of a strengthening of the extensionality crite­ rion for set equality which roughly states that two sets are equal if they are isomorphic in a suitable sense. This gives rise to an anti-foundation axiom that we call FA FA. It turns out that there is an alternative notion of isomorphism between sets which gives rise to yet another anti-foundation axiom which we call SAFA . In all we consider the four specific anti-foundation axioms AFA , BAFA, FA FA and SAFA. Each can b e consistently added t o the axiom system ZFC- and each gives rise to an axiom system in which every possible non-well-founded set exists when account is taken of the particular criterion of set equality that is being used. Nevertheless the four axiom systems are incomparable in the sense that in ZFC- no one of the four axioms AFA , SAFA , FA FA or BAFA can be proved from any other, assuming that ZFC- is consistent . Each of the four anti-foundation axioms was first formulated in one way or another by someone else. Nevertheless here I at­ tempt to consider them all in a uniform setting. So I have chosen to introduce my own more uniform terminology. The reader should examine the Notes towards a History, at the end of the book, to find out out more about the earlier work. The original stimulus for my own interest in the notion of a non-well-founded set came from a reading of the work of Robin Milner in connection with his development of a mathematical the­ ory of concurrent processes. This topic in theoretical computer science is one of a number of such topics that are generating exciting new ideas and intuitions that are in need of suitable mathematical expression. In chapter 8 I outline how I see the rv

rv

xx

Introduction

relationship between Milner's ideas and the axiom A F A and non­ well-founded sets. Another major area of application for the notion of a non­ well-founded set and the axiom AFA is to situation theory. Jon Barwise realised the significance of A FA for situation theory while I was giving the lectures that form the origin of this book. I have chosen not to present any of the details of this application here. Instead, in chapters 6 and 7, I have focussed attention on what I consider to be some of the fundamental general mathematical ideas that are being exploited when using A FA . Some of the ideas and terminology have been presented in an elegant and appealing way in the book The Liar, by Jon Barwise and John Etchemendy, and I have taken the opportunity to incorporate those ideas into this book.

Part One

The Ant i-Foundat ion Axiom

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1

I

Introducing the Axiom

Pictures Of Sets

Sets may be pictured using ( downward growing ) trees. For ex­ ample if we use the standard set theoretical representation of the natural numbers , where the natural number n is represented as the set of natural numbers less than n, then we have the following pictures for the first few natural numbers: o •

1

3

I

More generally pointed graphs may be used as pictures of sets. For example we have the following alternative pictures of 2 and 3: 3

A So what exactly is a picture of a set? We need some termi­ nology. Here a GRAPH will consist of a set of NODES and a set 3

4

The Ant i-Foundat ion Axiom

of EDGES, each edge being an ordered p air ( n, n' ) of nodes. If ( n, n' ) is an edge then I will wr ite n --> n' and say t hat n' is a CHILD of n. A PATH is a finite or infinite sequence --> . . . of nodes no, nl, n2, ... linked by edges (no, nl), (nl, n2 ) , . . . . A POINTED GRAPH is a graph together with a distinguished node called its POINT. A pointed graph is ACCESSIBLE if for every node n there is a path no --> nl --+ ... --> n from the point no to the node n. If this path is always unique then the pointed graph is a TREE and the point is the ROOT of the tree. We will use accessible pointed graphs ( apgs for short ) as our pictures. In the diagrams the point will always be located at the top. A DECORATION of a graph is an assignment of a set to each node of the graph in such a way that the elements of the set as si gn ed to a node are the sets assigned to the children of that node. A PICTURE of a set is an apg which has a decoration in which the set is assigned to the point . Notice that in our example s there i s only one way t o decorate the apgs. For example the last diagram must be decorated in the following way. no

-->

n1

--->

n2

The node labelled 0 has no children and hence must be assigned the empty set, i.e. 0, in any decoration. The central node has as only child the node labelled o. Hence in any decoration the central node must be assigned the set { O } , i.e. 1 . Continuing in this way we are inevitably led to the above decoration, and this decoration shows us that we have a picture of 3. Reflecting on how t he decoration was formed we are led to a formulation of an important result of set theory. Call a graph WELL-FOUNDED if it has no infinite path. Mostowski's Collapsing Lemma: Every well-founded graph has a unique decoration.

Introducing the Axiom

5

This result is proved by a simple application of definition by recursion on a well-founded relation to obtain the unique function d defined so that dn = {dn' I n ---> n' } for each node n of the graph. The decoration d assigns the set to the node n. Note the obvious consequence.

dn

1.1 Corollary:

Every well-founded apg is a picture of a unique set.

Which sets have pictures? There is a simple answer to this question. 1.2 Proposition:

Every set bas a picture.

To see this we will associate with each set a its CANONICAL PICTURE. Form the graph that has as its nodes those sets that occur in sequences ao , aI, a 2, . . . such that

... E

a2

E

al

E

ao = a

and having as edges those pairs of nodes (x, y) such that y Ex. If a is chosen as the point we obtain an apg. This apg is clearly a picture of a, the decoration consisting of the assignment of the set x to each node x. Note that this construction does not require the set a to be well-founded . Every picture of a set can be unfolded into a tree picture of the same set . Given an apg we may form the tree whose nodes are the finite paths of the apg that start from the point of the apg and whose edges are pairs of paths of the form (ao

--+

. ..

-- a , ao

--+

.,.

--+

a

--+

a

'

).

The root of this tree is the path ao of length one. This tree is the UNFOLDING of the apg. Any decoration of the apg induces a decoration of its unfolding by assigning to the node ao --+ . . . --+ a of the tree the set that is assigned to the node a of the apg by the decoration of the apg. Thus the unfolding of an apg will picture any set pictured by the apg. The unfolding of the canonical picture of a set will be called the CANONICAL TREE PI CTU RE of the set. Our discussion so far has been intended to motivate the fol­ lowing axiom:

The Anti-Foundat ion Axiom

6

The Anti-Foundation Axiom, AFA: Every graph has a unique decoration.

Note the following obvious consequences . • •

Every apg is a picture of a unique set . Non-well-founded sets exist .

In fact any non-well-founded apg will have to picture a non-well­ founded set. Examples of Non-Well-Founded Sets

In the rest of this section we will examine some pictures of non­ well-founded sets assuming the new axiom . Of course we must relinquish the foundation axiom, but it will turn out that we need drop none of the other axioms of set theory. 1 . 3 Example: Consider the apg

o This is a picture of the unique set 0 such that

0 = {O }

.

This is our first example of a non-well-founded set . When the apg above is unfolded we get the infinite tree

An analogous 'unfolding' of the equation above would seem to give us 0 = {{{- . } } } , .

Introducing the Axiom

7

if only the infinite expression on the right hand side had an in­ dependently determined meaning! The infinite tree above and the infinite exp ression associated with it might suggest that in some sense 0 is an infinite object . But a mome nt ' s thought should convince the reader that 0 is as finite an object as one could wish. After all it does have a finite picture. We may call set s that have fi ni t e p ict ur es HEREDITARILY FINITE sets. o has many terisation.

pi ctures. In fact we have the following charac­

1 .4 Proposition: An apg is a picture of 0 if and only if every node of the apg has a child. Proof: Assume given a picture of 0 with root a. Let d be a decoration of the p i cture such that da = o. Now if b is any node of t he picture there must be a pat h a = ao ----> . ----> an = b so that db = da n E . . . E dao = da = O. As 0 is the only element of 0 it follows that db = O. As 0 has an element it follows that b must have a child. Thus every node of the picture must have a child. Conversely assume given an apg with the property that every node has a child. Then the as signment of 0 to each node of the apg is easily seen to be a decoration of the apg, so that the apg is a picture of O . 0 .

.

1 . 5 Example: The apg

is a picture of the unique set 0* such that

0*

=

{a, O*}.

W hen "unfolded" this equation becomes

0* = {a, {a, {O, . . . }}} . 1.6

Example: We have seen that every set has a picture. Let

8

The Anti-Foundation Axiom

denote a picture of some set a. Then

is a picture of the unique set a* such that

a*={a,a*}. we get the special case in example 1.5. Again the above equation can be 'unfolded' in the obvious way.

If a

=

°

Let us now consider the special case when a =O. 0* is the unique set such that 0 * = {O,O*}. But 0 = {O} = {O,O}. Hence we must conclude that 0* =O. Of course this is also clear from the characterisation of pictures of 0 given earlier. 1 . 7 Example: The ordered pair of two sets is usually repre­ sented as follows:

(a, b) = {{a},{a, b}}. So the equation

x= (O,x) becomes

x = {{O}, {O,x}}. This equation in one variable x is equivalent in an obvious sense, to the following system of four equations in the four vari­ ables x, y, z, w.

x= {y, z} y= {w} z={w,x} w=O

Introducing t he Axiom

9

Now these equations hold exactly when the following diagram is of a correctly decorated apg.

x

w

Hence by AFA the above system of four equations has a unique solution and hence the original equation x = ( O,x)

has a unique solution with picture

"Unfolding" this eq u at i on

we

get

x = ( 0, (0, (0, . .. ))) 1.8 Example: As in example 1.6 the previous example may be generalised to show that for any set a the equation x = ( a, x) has a unique solution x = ( a, ( a, ( a, ... ))) . More generally still, given any infinite sequence of sets ao, aI, a2, ... we may consider the following infinite system of equations

10

The Anti-Foundation Axiom

in the variables xo, Xl, X2, X3,···

Xo = (ao, xI) Xl = (aI, X2) X2 = (a2, X3)

It should be a straightforward exercise for the reader to show that this system of equations has a unique solution. Infinite ex­ pressions for this unique solution can be obtained by 'unfolding' the system of equations to get

Xo = Xl = X2 =

(ao, (aI, (a2," . ))) (aI, (a2, (a3, . ..))) (a2' (a3, (a4,...)))

This and other examples can be treated even more simply using the following strengthening of AFA . A LABELLED GRAPH is a graph together with an assignment of a set al of LABELS to each node a. A LABELLED DECORATION of a labelled graph is an assign­ ment d of a set da to each node a such that

da = {d b I a

-t

b} U al .

The Labelled Anti-Foundation Axiom: Every labelled graph has a unique labelled decoration.

The ordinary anti-foundation axiom may be viewed as a spe­ cial case by treating ordinary graphs as labelled graphs with an empty set of labels attached to each node. Conversely, we will see that the labelled anti-foundation axiom is a consequence of the ordinary one.

Introducing the Axiom

11

Now given sets ao, aI, a2, . . . we can obtain the sets X n such that X n = (an , x n +d for n = 0, 1, . . . in the following way. Con­ sider the labelled graph having natural numbers as nodes, an edge n --. n + 1 for each n and sets of labels given by: (2n)1

=

{ {an } } ,

(2n + 1)1

=

{ an } .

Using the labelled anti-foundation axiom let d be the unique la­ belled decoration of the labelled graph. Then for n = 0, 1, ...

Hence if Xn

=

d(2n)

=

d(2n + 1 )

=

{d(2n + I) } U { {an } } , {d(2n + 2) } U {an } .

d(2n) then Xn =

{d(2n + 1 ) , {an }}

= { { x n +l, an } , {an }}

for each n. Hence we have obtained the desired sets X n . Their uniqueness easily follows from the uniqueness of d. There is an even more powerful technique that can be used to deal with this and other examples. This technique involves the formulation of a result asserting that every system of equations of a certain type has a unique solution. We can then simply apply the result directly to each example without the need for any coding. Following the terminology of Barwise and Etchemendy the result will be called the solution lemma below. In order to formulate the lemma in an intuitively appealing way we need to consider an expansion of the universe of pure sets that we have been considering so far. Pure sets can only have sets as elements and those sets are also pure. The expansion of the universe involves the addition of atoms and sets built out of them. Atoms are objects that are not sets and are not made up of sets in any way, so that they have no set theoretical structure. But they can be used in the formation of sets. See ( Barwise 1975) for a discussion of the formalisation of set theory with atoms. In that book atoms are called Urelemente. The construction of an expanded universe by adj oining to a universe of sets atoms and adding all the sets that can involve these atoms in their build

12

The Ant i-Foundation Axiom

up is analogous to the construction of a polynomial ring from a ring by adjoining indeterminates and adding all the polynomials

in those indeterminates with coefficients taken from the ring. It will be convenient to assume that we have a plentiful supply of atoms. So we assume that for each pure set i there is an atom Xi, with Xi f:: Xj for distinct pure sets i, j. If X is a class of atoms then we call sets that may involve atoms from the class X in their build up X-SETS. The solution lemma will apply to a system of equations of the form (X E X ) , where ax i s an X-set for each X E X . For example given pure sets ao, al, . . . the system of equations

(n = O, l , .. . ) has the above form when we take X = {xo, Xl, . . . } and for each n we take aXn to be (an' xn+d; i. e the X-set {{an}, {an, xn+d}. In this example it is clear what a solution of this system of equations must be. It is a family of p ure sets bo, bl, , one for each atom in X, s uch that .

.

bn

=

(an, bn+d for

n

=

.

.

0,1, ....

Notice that the right hand sides of these equations are obtained from the right hand sides of the original system of equations by substituting bn for each atom Xn. Th is suggests what a solution to the general system of equations should be. It should be a family 7r = (bX)XEX of pure sets bx, one for each X E X, such that for each x E X bx = 7rax· Here, for each X -set a, the set tra is that pure set that is obtained from a by substituting bx for each occurence of an atom X in the build up of a. So 7r is the substitution operation characterised in the following res ult .

Substitution Lemma: For each family of pure sets 7r = (bx)XEX there is a unique oper­ ation 7r tha t assigns a pure set tra to each X -set a such tha t

n-a

=

{n-b I b is an X-set such that b E a}

U

{7rX I X E a n X}

.

Introducing t he Axiom

13

We can now state the result we have been aiming at . Solution Lemma: If ax is an X -set for each atom x in the class X of atoms then the system of equations x

=

ax

(x E X)

has a unique solution; i.e. a unique family of pure sets 7r (bx)xEX such that for each x E X

The above informal discusion of the solution lemma seems to be all that is required when seeking to apply the lemma. A rigorous formulation and proof will be left till the end of this chapter. Systems

We need to widen the notion of a graph so as to allow there to be a proper class of nodes. A SYSTEM is a cl as s M of nodes together with a class of EDGES consisting of ordered pairs of nodes. We shall s imp ly use M to refer to the system and write that a --+ b in M or simply a --+ b if (a, b) is an edge of M. A system M is required to satisfy the condition that for each node a the class aM = {b E M I a --+ b} of children of a is a set . Note that a graph is simply a small system. An example of a l arge system is the universe V with a --+ b whenever b E a. The notion of a decoration of a graph extends to systems in the obvious way. We get the following strengthening of A FA. 1 .9 Theorem: (assuming AFA) Each system has a unique de corati on

.

Proof: Let M be a system. To each a E M we may associate an apg Ma const ructed as follows: •

The nodes and edges of M a are those nodes and edges of M that lie on paths of M starting from the node a, and the point of M a is the node a itself.

That the nodes of M a do form a set may be seen as follows. Let = {a} and for each natural number n let

Xo

14

The Anti-Foundat ion Axiom

Xn+l = U{XM I x E Xn} . i s a set for all x E M , each Xn is a set , the set of those nodes of M that end paths in M of length n starting from the node a. So the nodes of M a form the set Un Xn. By AFA each apg M a has a unique d ec or ation da so that M a will be a picture of the set daa. For each a E M let da = daa. We will show that d is the unique decoration of M. First observe that if a -+ x in M then every node of M x will also be a node of M a and the restriction of da to M x will be a decoration of M x and hence equal to dx , the unique decoration of M x . Hence if a -+ x in M then dax = dxx = dx. So, for each a E M, As

XM

da = daa {dax I a -+ x in M a } {dx I a x in M}. Thus d i s a decoration o f M. To see the uniqueness of this deco­ =

=

-+

ration it suffices to observe that any decoration of .A1 must be a decoration of each M a, when restricted, and hence must extend each da, so t hat it must be d itself. 0 LABELLED SYSTEMS an d their labelled decorations are defined in the obvious way. If a E M then alNf will denote the set of labels at a in the labelled system M. We next generalise the previous result to labelled systems. 1.10

Theorem: (assuming AFA) Each labelled system has a unique labelled decoration.

Proof: Let M be a labelled system. Let M' be the system having (i, a) such that either i = 1 and having as edges:

as nodes all the ordered pairs a E M or i = 2 and a E V and •

•

•

By

(1, a) (1, a ) (2, a )

-+ -+ -+

AFA M'

( 1 , b) whenever a -+ b in M, (2, b) whenever a E M an d b E al M, (2, b) whenever b E a.

has

a unique decoration

1r(l,a)

=

{1r(l,b) I a

and for each

a

E

-+

7r.

So for each a E M

b in M } U {1r(2,b) I b E alM}

V

1r(2,a)

=

{1r(2,b) I b E a}.

Introducing t he Axiom

15

Note that the assigment of t h e set 7r(2, a) to each a E V is a decoration of the system V so that by AFA 7r(2, a) = a for all a E V. Hence if we let ra = 7r(1, a) for a E M then, for a E M , ra

=

{rb I a

b in M} U alM,

-+

so that r is a lab e lled decoration of the labelled system M. For the uniqueness of r suppose that r' is a labelled deco­ ration of the labelled sy stem M. Then 7r' is a decoration of the system M' where 7r' (1, a)

= r'

7r' (2, a )

=

It follows from AFA that 7r' r'a and hence r'

=

a for a E M, for a E

a

=

7r'(l,a)

V.

7r so that for a E M =

7r(l,a)

=

ra, o

= r.

We next give a ge n er al result that will then be used to prove the Substitution and Solution Lemmas. 1.11

Theorem: (assuming AFA) Let M be a labelled system sets of labels are subsets of the class X.

whose

(1) If 7r : X -+ V then there is a unique map ir that for each a E M

ira

=

{irb I a

-+

b in M} U {7rX I x

E

:

M

-+

V

alM}.

(2) Given ax E M for x E X there is a unique map 7r : X such that for all x E X 7rX

=

such

-+

V

irax.

Proof: (1) For each 7r : X -+ V let Mrr be the labelled system obtained from M by redefining the sets of labels so that for each node a alMrr = {7rX I x E alM}. Then the required unique map oration of Mrr.

ir

is the unique labelled dec­

16

The Anti- Foundation A xiom

(2) Let M' be the system having the same nodes as M, and all the edges of M together with edges a ---+ ax whenever a E M and x E al M . By theorem 1.9 M' has a unique decoration

a

a,

23

bE M

b.

a, b

EM

= b.

2.9 Exercise: Show that

AFA2 AFA2xt ,

where AFA2xt is: Every extensional graph has at most one decoration. Observe that by (i) of exercise 2.8 every strongly extensional system is extensional . Note that by the extensionality axiom the system V is extensional. By the next result AFA2 expresses a strengthening of the extensionality axiom. 2 . 10 Proposition: AFA2 V is strongly extensional.

Proof: Let us first assume AFA2 and let a ==V b. Then by exercise 2.5 a == b so that there is an apg Gn and decorations dl and d2 of G such that d1 n = a and d2n = b. By AFA2 d1 = d2 so that a = b. Thus V is strongly extensional. Conversely let V be strongly extensional and let dl and d2 be decorations of a graph G. If x E G then dlX == d2X , as Gx is a picture of both dl X and d2X. Hence, by exercise 2.5 dl X ==V d2X, so that dl X = d2X, as V is strongly extensional. Thus dl = d2 so that we have proved AFA2. 0 System Maps

A SYSTEM MAP from the system M to the system M ' is a map 1r : M --+ M' such that for a E M If

1r

is a bijection then it is a SYSTEM ISOMORPHISM.

2 . 1 1 Example: A system map G

simply a decoration of the graph.

--+

V, where

G

is a graph, is

2 . 1 2 Exercise: Show that systems and system maps form a

(superlarge) category.

24

The Anti-Foundation Axiom

The close relationship which exists between bisimulations and system maps is illustrated by the following results. 2 . 1 3 Proposition: Let 71"1 , 1T2 : M

--+

M' be system maps.

( 1 ) If R is a bisimulation on M then

is a bisimulation on M' . (2) If S is a bisimulation on M' then

is a bisimulation on

M.

Proof:

( 1 ) Let S = (1Tl x 1T2 )R and let bl Sb2 and b� E b l M, . Then there are aI , a2 such that al Ra2 and bl = 1Tl al , b2 = 1T2a2 . As b� E (1Tad M' there is a� E (ad M' such that b� = 1Ta� . As R is a bisimulation on M there is a; E (a2 ) M such that a� Ra; . Now if b; = 71"a; then b� Sb; and b; E ( b2 ) M . Thus we have '

proved that if b l Sb2 then

Vb�

E

( b l ) M ' :3b; E (b2 ) M' b� Sb; .

Similarly, if bl Sb2 then also

Thus S is a bisimulation on M'. (2) Let R = (71"1 X 1T2 ) - I S and let al Ra2 and a� E aM . Then (1Tl ad S(71"2a2 ) and 1Tl a� E (1Tl al ) M' . As S is a bisimulation on M' there is b; E (71"2 a2)M' such that (1T l aDSb; . So b; = 71"2a; for some a; E (a2 ) M . So a� Ra; for some a; E (a2 )M . Thus if a1 Ra2 then

and similarly we get o

The Axiom in More Detail

25

2 . 14 Exercise: Let R be a bisimulation on M. Let Mo be the system whose nodes are the ordered pairs in the relation R with

(a, b)

--+

Let 7r1 , 7r2 : Mo

(a' , b') in Mo --+

iff a

a' and b

--+

--+

b' in M.

M be given by 7r1 (a, b) = a, 7r2 (a, b) = b,

for a, b E M . Show that 7r1 and 7r2 are system maps. This exercise generalises a construction in the proof of propo­ sition 2.5 to an arbitrary system M. 2 . 1 5 Exercise: If M is a system show that a =M b if and only if there is a graph G and system maps dl , d2 : G --+ M such that a = dI n and b = d2 n for some n E G .

This exercise generalises proposition 2.5 and was suggested by D ag Westerstahl. Let 7r : M --+ M' be a quotient of the class M with respect to the equivalence relation R on M ; i.e. 7r is a surjective map such that for all a, b E M

aRb

�

tra = 7rb.

Now suppose that M is a system and R is a bisimulation on M. Then 7r is a system map, provided that M' is made into a system by letting the edges of M' be all the pairs (7ra, 7rb) for (a, b) an edge of M. We will say that 7r : M --+ M' ( or sometimes, simply M') is a QU OTIENT of the system M with respect to the bisimulation equivalence R. Note t hat any two such quotients will be isomorphic systems.

M' is a quotient of the system M with respect to a bisimulation equivalence R then M' is strongly extensional if and only if R is the relation = M . 2 . 1 6 Exercise: Show that if 7r : M

--+

7r : M --+ M' is a quotient of the system M with respect to = M then we say that it is a ST RONGLY EXTENSIONAL QUOTIENT of M. If

2.17 Lemma: Every system has a strongly extensional quotient.

26

The Anti-Foundation Axiom

Proof: The essential problem is to define a map the system M such that for aI , a2 E M

1r

with domain

For small M the standard definition of 1r in terms of equiva­ lence classes would work. In general a strong form of global choice would be needed to pick a representative from each equivalence class. Here we shall g ive an argument that only uses the local form of AC. For each a E M the set of nodes of the apg M a is in one-one correspondence with an ordinal, an d the correspondence induces an apg s truct ure on the ordinal. The resulting apg will be in the universe of well-founded sets and will be isomo rphi c to M a. For each a E M let Ta b e the class of apgs in the well­ founded universe that are isomorphic to Ma' for some a' E M su ch that a =M a' . By the above each class Ta is non- empty and hence has elements of minimum possible rank in the well-founded universe. Let 1ra be t he set of such elements of Ta . Note that if al =M a2 then Tal = Ta 2 so that 1ral = 1ra2 . C onversely if aI , a2 E M such that 1ral = 1ra2 then there must be an apg in both Ta l and Ta2 • Hence there must be a� , a2 E M such that - , al '" M a2' ' B y exerCl's e 2 . 8 a 'l = - M a ,2 =M aI, , a 2 = M a2 and M a 'l = so that al =M a2 · 0 2 . 1 8 Exercise: (due to

Dag

A FAI

Westerstabl) Sbow tbat



a

=

b.

4.2 Exercise: Show that if M is ",-extensional then for any system Mo there is at most one injective system map Mo ----t M . 41

42

Variants of the A nti- Foundation Axiom

Hint: Observe that if 7r : Ml then for a E Ml

----t

M2 is an injective system map

/"V - Complete Systems

A system M is a "' COMPLETE system if it is ",-extensional and every ",-extensional graph has an M -decoration. Note that by exercise 4.2 the M -decoration is necessarily unique. -

4.3 Example: If '" is •

•

= vo

then

M is ",-extensional iff M is strongly extensional . M is "'-complete iff M is complete .

Our first aim is to construct a ,,-,- complete system. Let Vo� be the subsystem of Vo consisting of the ,,-, extension al apg's and all the edges of Vo between such apg's . We let Vc� be a "'­ extensi onal system for which there is a surjective system map 7r� : Vo� ----t Vc� such that -

Ca '"

C'a' ; �

7r(Ga)

=

7r(C'a')

for all ",, - extensional apg's Ga and C' a' . We are gu ar anteed the existence of v::� and 7r� by the following lemma. 4.4 Lemma: For every system M there is a system M' and surjective system map 7r : M ----t M' such that for x , x' E M

7rX

Mx '" Mx'

=

7rX , .

Moreover if M x is ",-extensional for all x E M then M' extensional.

IS "'­

Prool The first part is proved as in the proof of lemma 2. 17. For the second part observe that for each a E M the restriction of 7r to M a is a surj ective system map

Ma

----t

M ' ( 7r a) .

that x, y E Ma and 7rX = 7ry . Then Mx '" My so that by the "'-extensionality of Ma it follows that x = y. Thus 7r r M a is an isomorphism M a � M'(7ra) so that M a "" M' (7ra) . Now suppose

Vari ants Using a Regular Bisimulat ion

We can now show that M' is ,,-,-extensional. As 7r : M is surjective it suffices to show that ==>

M ' (7ra) "-' M ' (7rb)

tra

=

--

43

M'

7rb.

But assuming that M' (7ra) "-' M' (7rb) we get by the above that Ma "-' M ' (7ra) "-' M ' (7rb) "-' Mb, so that M a "-' Mb and hence tra

=

7rb.

o

Note that in applying this lemma to M = Vo� , if x = Ga E M then M x � Ga so that M x "-' Ga and M x is ,,-,-extensional, as Ga is. 4.5 Proposition: For each ,,-,-extensional system M there is a unique injective system map M ---+ V';;� . Proof: B y exercise 4 . 2 the uniqueness of the Injective system map follows from the fact that Vc� is ,,-,-extensional. So it only remains to show the existence of a system map M -- V';;� , where M is ,,-,-extensional. Clearly 7r M : M -- Vo� is a system map, where 7rM a = Ma for a E M . Hence by composing with 7r� : Vo� -- Vc� we get a system map 7r� 0 7rM : M ---+ V';;� . It only remains to show that this map is injective. So let x, y E M such that 7r� (Mx) = 7r� (My) . Then Mx "-' My so that , as M is ,,-,-extensional, x = y . o 4.6 Corollary: V';;� is ,,-,-complete. Proof: We already know that V';;� is ,,-,-extensional. It only re­ 0 mains to apply the proposition to small systems M.

For systems M, M' let M � M' if there is an injective system map M ---+ M ' . Note that � is both reflexive and transitive. Our next aim is to prove the following result . 4.7 Theorem: Let M be a ,,-,-extensional system. Then the

following are equivalent:

( 1 ) M is ,,-,-complete. (2) Mo � M for every ,,-,-extensional system Mo . (3 ) M � M' ==> M � M' for every ,,-,-extensional system M' . (4) M � Vc� .

44

Variants of t he A nt i-Foundation Axiom

Proof: For ( 1 ) implies ( 2 ) let M be "'-complete and let Mo be a ",-extensional system. Then for a E Mo the apg Moa must have an injective M -decoration da which, by exercise 4 . 2 , is uniquely determined. Define d : Mo --t M by da

=

daa for a E

Observe that if a E M then dx ( daa)M

=

=

Mo.

da z ( Mox ) for x E aM , so that

{ dxx I x E a Mo } ,

and hence (da)M = {dx I x E aMo } ' Thus d is a system map. To see that d is inj ective use the hint to exercise 4 . 2 to get that dx : Mox

�

M ( dx) and dy : Moy

�

M (dy ) ,

so that if dx = dy then Mox � Moy. It follows that if dx = dy then Mox '" Moy and hence x = y, as Mo is "'-extensional. For ( 2) implies (3) let M :5 M', where M' is "'-extensional . By (2) M' � M . So there are injective system maps M --t M' and M' --t M. By exercise 4 . 2 their compositions must be the identity maps on M and M' so that M � M' . For (3) implies (4) use proposition 4.5 to get that M :5 "V;;� . As "V;;� is ",-extensional we may apply (3) to get (4) . For (4) i mplies ( 1 ) apply Corollary 4 . 6 . 0 The next result generalises proposition 3.7. 4.8 Lemma: Every rv-complete system is full.

Proof: Let x � M be a set, where M is a "'-complete system. As in the proof of proposition 3.7, we may form the graph Go co nsis ting of the nodes and edges o f M that lie on paths starting from a node in x. Again we may let G be obtained from Go by adding a new node * and edges ( * , y) for y E x . If G is ­ extensional then by taking the unique M -decoration of G we can argue as before. But if G is not rv-extensional then, as Go is, it must be the c as e that G* Ga for some a E Go . As rv is a bimulation it follows that "'

'"

Vy E *G3a' E aa

Gy

rv

Ga'

&

Va' E aG 3 y E *G Gy

But *G = x and a E M with aG = aM · Also and Ga ' = Ma' for a ' E aGo Thus

Vy

E

x3a' E

aM

My

rv

Ma'

&

Va'

E

Gy

=

aM3y E

x

rv

Ga' .

My for y My

rv

E

x

Ma' .

Variants Using a Regular Bisimulation

Hence,

as

45

M is '"'-'-extensional x = aM .

The uniqueness of a is a consequence of the fact that M is '"'-'-extensional and hence extensional. 0 The Axioms

A FA'"

So far we have not given our generalisation of A FA . To do so we must assume given a definition in the language of set theory of the regular bisimulation "'. So we assume given a formula ¢( x , y) , without any parameters and having at most the variables x, y free, that defines '" in V. This means that for all apg's c and d c

'"'-' d

�

V F ¢( c, d) .

We shall assume that ¢(x, y) is fixed and refer to it as the defini­ tion of "'. Using the definition of '" we may form a sentence that expresses that V is '"'-'-complete. Let us call this sentence A FA� . It is this sentence that is our generalisation of A FA . Note that in case '"'-' is = vo then AFA�

�

A FA .

A s with A FA we may split A FA� into its two parts • •

A FA1 : Every ",-extensional graph has an injective decora­ tion . AFA2 : V is ",-extensional. The following is straightforward to prove.

4.9 Proposition: ( 1 ) A FAI iff every '"'-'-extensional apg is an exact picture. (2) A FA2 iff every exact picture is ",-extensional. 4.10 Corollary; A FA� is equivalent to: an apg is an exact picture iff it is ",-extensional.

A result that we have not yet generalised is theorem 3 . 8; i.e. t hat each complete system M i s a full model of ZFC - + A FA . To do so we need to assume that the definition of '" is absolute for full systems . To spell out what this means let M be a full system

46

Variants of the Anti-Foundation Axiom

and let M be the relation on M th at the definition ¢( x, y) of defines in M; i.e. for c, d E M '"

C

¢=::}



a = b.

But on examining Finsler 's paper this is clearly seen to be incor­ rect. In fact Finsler understands his axiom II to be a strength­ ening of the extensionality axiom. But e:: - extensional systems need not be extensional. For example consider the two element graph G:

It has nodes a and b and edges (a, b) and (b, b) . Clearly Ga � Gb but aa = {b} = ba . So G is e:: -extensional but not extensional. A correct formulation of Finsler's notion of isomorphism will be given using the following construction. If a E M, where M is a system, let (M a) * be the apg consisting of the nodes and edges of M a that are on paths starting from some child of a, together with a new node * and a new edge ( * , x ) for each child x of a. We take * to be the point of ( Ma) * . Note that if a does not lie on any path starting from a child of a then (M a)* will be isomorphic to M a via an isomorphism that is the identity except that * is mapped to a. If a does lie on such a path then ( M a) * consists of the nodes and edges of M a together with the new nodes and edges. We define a, b E M to be isomorphic in Finsler's sense if (Ma)* e:: (Mb) * . Note that if aM = bM then (Ma)* = (Mb) * and hence (M a)* � (Mb)* .

48

Variants of t he Ant i-Foundation Axiom

Let �* be the relation on Vo defined by:

Ga �* G' a'

�

(Ga)* � (C' a'r .

We call a system M a FINSLER-EXTENSIONAL system if it is �*­ extensional; i.e.

Ma �* Mb

�

a

=

b.

It is the Finsler-extensional systems that we take to be the models of axiom II. 4. 13 Exercise: Show that ( i ) �* is a regular bisimulation. ( ii ) A system M is Finsler-extensional iff it is both extensional

and �-extensiona1.

4.14 Exercise: Let "" be the relation on Vo : C a "" G' a' iff there is a bijection '!jJ : aG � aCI such that Gx � G' ('!jJx) for x E aG o

Show that

(i) Ga � * C'a' � Ga '" C'a'. (ii) is a regular bisimulation. (iii) M is ",, - extensional iff M is Finsler-extensional. '"

Let us now consider Finsler's axiom III. I take a Finsler­ extensional system to be a model of axiom III if any i nje c tive system map M --- M' is an isomorphism if M' is a Finsler­ extensional system. By theorem 4.7 a system M is a model of Finsler's axioms iff M is Finsler-complete ( Le. M is �* -complete) . 4 . 1 5 Exercise: Show that � * has a definition that is absolute

for full systems.

By this result we may form the axiom A FA�' , which we FINSLER'S ANTI-FOUNDATION AXIOM , or FA FA for short . The previous work applies to give us the fol lowin g two results. will call

4. 1 6 Theorem: FAFA is equivalent to: An apg is an exact picture iff it is Finsler- extensional. 4.17

up to

Theorem: ZFC- + FA FA has a full model that is unique

isomorphism.

Variants Using a Regular B isimulat ion

49

Scott 's Anti-Foundation Axiom

In Scott ( 1 960) a model of ZFC- with non-well-founded sets is constructed out of irredundant trees . Scott defines a tree to be a REDUNDANT tree if it has a proper automorphism; i .e. an auto­ morphism that moves some node. The tree is an IRREDUNDANT tree otherwise. Scott ( 1960) gives another characterisation of this notion. We leave this as an exercise. 4. 18 Exercise: Show that a tree Tr is redundant iff there is a node c of Tr and distinct a, b E CT such that T a � Tb.

Scott 's idea is to use irredundant trees to represent the struc­ ture of sets. Recall that the canonical tree picture of a set c is obtained by unfolding the canonical picture Vc of c. Scott's model construction may be described as follows. Let Vci be the subsys­ tem of Va consisting of the irredundant trees with all the edges of Va between such nodes . A system vet and a surjective system map 7r : VJ ---t vet are constructed so that for trees Tr, T'r'

7r ( Tr )

=

7r ( T' r ' )

{=}

Tr � T' r' .

�t can be shown to be full and hence a model of ZFC- . Moreover it is also a model of

A tree is isomorphic to a canonical tree picture iff it is irre­ dundant . We shall call this SCOTT ' S ANTI-FOUNDATION AXIO M , or SA FA for short. It is essentially the axiom formulated in Scott ( 1 960) . In the rest of this section we will show that the axiom SA FA and its full model �t are really special cases of the axiom A FA� and its model Vt for a suitable choice of the regular bisimulation "'. For any apg Ga let (Ga)t denote its unfolding. So the nodes of (Ga)t are the finite paths of Ga that start from a. Let �t be the relation on Va given by •

Ga �t G'a'

{=}

(Ga)t � (G'a' ) t .

4 . 1 9 Exercise: Show that � t is a regular bisimulation which

has a definition that is absolute for full models. By this result we may obtain the axiom A FA� t and its model ��t . The next three results will be needed to show that AFA� t is equivalent to SAFA.

50

Variants of the Anti- Foundation Axiom

4.20 Lemma:

The unfolding of a � t -extensional apg is an irredundant tree.

Proof: Let Gn be a �t-extensional apg. Let a, b E

ce , where E (Gn)t , such that (Gn)ta � (Gn)t b. Then (G a ) t = (Gn) t a � ( Gn ) t b = ( Gb) t so that Ga � t Gb and hence a = b as G is � t _ extensional. Thus (Gn)t is irredundant . 0 c

4.21 Lemma: If T r is an irredundant tree then there is a ':::::. t extensional apg Gn and a surjective system map 7f : Tr --+ Gn such that Tr � (Gn)t and for a, b E Tr 7fa =

7fb

Ta

�

�

Tb.

Proof: Let Tr be an irredundant tree. Let '" be the equivalence relation on the nodes of Tr defined by a '" b

�

Ta

�

Tb,

for a, b E Tr. As '" is a bisimulation equivalence we can form a quotient 7f : Tr ---+ Gn of Tr with respect to '" by letting 7fa =

{b E T r I

a

'"

b}

for a E Tr , and letting G = {7fa I a E Tr} and n = 7fT . It only remains to show that Tr � (Gn)t . So define 1j; : Tr ---+ (Gn)t by: 1j;a

=

(7fr, . . . , 7ra)

for a E Tr, where r ---+ . . . ---+ a is the unique path in Tr between the root r and the node a. That 1jJ is a subjective system map should be clear. To see that 1j; is inj ective let a, b E Tr such that 1j;a = 1j; b = (n, . . . , c ) . Then there are paths r ---+ ---+ a and . . , 7f a = 7r b = c . Suppose ---+ b in Tr such that 7fr = n, r ---+ that a i= b. Then there is a first node c in the path n ---+ ---+ c whose corresponding nodes a' and b' in the paths r ---+ ---+ a ' and r ---+ ---+ b are distinct, even though 7fa' = 7fb' = c . Then Ta ' � Tb' and a' and b' are children of the common node that precedes them in the paths r ---+ . . . ---+ a' ---+ ---+ a and r ---+ ---+ b' ---+ ---+ b. So, as T r is irredundant a' = b' , contradicting the choice of a' and b' . Hence we must have a = b . Thus 1j; is injective and hence 1j; : Tr � (Gn r 0 .

•

.

•

•

•

.

.

.

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

.

Variants Using a Regular Bisimulation

51

4.22 Lemma: If G a and G' a ' are 3:/ -extensional apg's then

Ga �t G' a '

==>

Ga � G' a ' .

Proof: Let Ga and G' a' be �t -extensional apg's such that Ga �t G'a' . Then by lemma 4.20 (Ga)t and (G'a')t are isomorphic ir­ redundant trees. Let 'IjJ : (Ga)t � (G' a' r Define 7r : Ga --+ G' a' as follows: If b E Ga let (]' be a path from a to b in Ga. Then (]' E (Ga)t so that 'IjJ(]' E (G'a')t . Let 7rb be the last node c in G' a' of the path 'IjJ(]' . To see that 7rb is well-defined let (]" also be a path from a to b in Ga and let c' be the last node in G ' a' of 'IjJ(]" . Observe that the subtrees of (Ga)t determined by the two paths (]' and (]" will both be isomorphic to the tree ( Gb)t and hence to each other. It follows that the corresponding subtrees of G' a' determined by 'I/J(]' and 'IjJ(]" will also be isomorphic. But these are isomorphic to (G'c)t and (G'c')t so that (G'c)t � (G'c,)t and hence G'c �t G ' c' . As G' is �t -extensional c = c' . Thus 7r is well-defined and a similar argument shows that 7r is injective. That 7r is also surj ective and is a system map should be routine to check. 0 The axiom SA FA may be split into the two parts: •

•

SAFAl : Every irredundant tree is isomorphic to a canonical tree picture. SA FA2 : Every canonical tree picture is irredundant .

4.23 Theorem:

( 1 ) SA FA2 (2) SA FA (3) SA FA

¢=> ==> ¢=>

AFA r · t A FA,="1 ==> t A FA ,=" .

SA FA l ·

Proof: First note that (3) is an immediate consequence of ( 1 ) and (2) . We now prove the four implications that make up ( 1 ) and

(2) . •

�t

SAFA2 ==> A FAz · Let a, b be sets and let c = {a, b} . Then by SA FA2 the tree (Vc)t is irredundant . As a, b determine children of the common node c of (Vc) t ( Va) t

�

(Vb)t

==>

a

=

b.

52

Variants of the Ant i - Foundation Axiom

•

•

•

Thus V is �t-extensional and so A FA r is proved . � t ==} SA FA2 . AFA= By A FA r the apg Va is �t-extensional so that by lemma 4. 20 the tree ( Va) t is irredundant . "" t

SA FA ==} A FA1 . Le t ea be a �t-extensional apg. Then by lemma 6 . 1 the tree (ea)t is irredundant . Hence by SA FA I there is a set c such that ( Ga ) t � (Vc) t . By ( 1 ) it follows from SA FA2 that Vc is �t-extensional. Hence by lemma 4 .22 Ga � Vc. Thus ea is an exact picture of c . "" t AFAI ==} SAFA1 . Let Tr be an irredundant tree and let 7r : Tr ---. Gn be as in lemma 4 . 2 1 so that en is �t-extensional and T � ( Gn)t . By A FA there is a set c such that en � Vc so that Tr � ( ea)t � ( Vc)t . Thus Tr is isomorphic to a canonical tree 0 picture .

t

4.24 Theorem: SA FA is equivalent to:

An apg is an exact picture iff it is Scott extensional. 4. 25 Theorem: ZFC- + SA FA has a full model that is unique

up to isomorphism.

The Relat ionship B etween t he A FA '" We

have considered three examples of regular bisimulations "" that have a definition that is absolute for full systems. In each case we get an axiom A FA� which has a unique full model up to isomorphism. The three relations are = vo , �* , �t and these determine the axioms A FA , FAFA and SA FA respectively. What is the relationship between these axioms? The following propo­ sition summarises what we know so far. The axioms A FA and FAFA are at opposite extremes of the family of axioms AFA� . While A FA expresses that only the strongly extensional apg's are exact pictures FAFA expresses that any Finsler-extensional apg is an exact picture. The axiom SA FA fits somewhere in between and it is the aim of this section to show that it fits strictly in between the two extremes. 4 . 26 Proposition: Let

be a regular bisimulation having a definition that is absolute for full models. Then rv

Variants Using a Regular B isimulation

53

(1) Every strongly extensional system is ,,-,-extensional. (2) Every ,,-,-extensional system is Finsler-extensional. (3) A FA 2 ==> A FA2' ==> FA FA2 . (4) FAFAI ==> A FAI ==> A FA1 . (5) If (a) : There is a ,,-,-extensional system that is not strongly extensional then -,( A FA1 & A FA 2 ) .

(6) If (b): There is a Finsler-extensional system that is not "-'­ extensional then

, ( FAFA 1 & A FA2' ) . (7) If both (a) and (b) then the axioms FA FA , AFA and A FA� are pairwise incompatible. 4.27 Theorem:

( 1 ) There is a � t -extensional graph that is not strongly exten­ sional. (2) There is a Finsler-extensional graph that is not �t -exten­ sional. Proof: ( 1 ) Consider the graph G:

with the distinct nodes a, b. This is �t-extensional because (Ga) t '1 (Gbr In fact (Ga)t is

54

Variants of the Anti-Foundation Axiom

while (Gb)t is simply

But G is clearly not strongly extensional. Note that assum­ ing A FA , Ga is a non-exact picture of 0 and Gb is an exact picture of O. On the other hand if we assume SA FA then Gb is still an exact picture of 0 but Ga is an exact picture of a set T t= 0 such that T = {O , T} . (2) This time let G be the graph : b

a

W c

with the distinct

of the

apg Ga

nodes a , b, c. Note that the unfolding the diagram :

has

tree have been labelled with the corresponding nodes of G. It is clear from this

where the nodes of the names of t he

(GaY

Variants Using a Regular Bisimulat ion

55

diagram that the subtrees (Gb) t and (Gc) t are isomorphic. This shows that G is not �t-extensional. But G is clearly extensional. Also it is rigid in the sense that it only has the identity automorphism. As every node is accessible from every other node it follows that G is Finsler-extensional. Note that assuming A FA the unique decoration of G will assign the set n to every node. But when SAFA is assumed there is a decoration of G in which the nodes b and c get assigned a set X and the node a gets assigned a distinct set Y such that Y = {X} and X = {X, Y } . Finally if FA FA is assumed there is a unique injective decoration that assigns pairwise distinct sets A, B, C to the nodes a, b, c respectively 0 so that A = {C} , B = {A, C} and C = { B , C} . When this theorem was presented in the course I only had an infinite example for part 2 . The first finite example was found by Randoll Dougherty after I raised the problem in a talk at Berkeley. His graph had 9 nodes and 26 edges and after a series of improvements the above simple example with only 3 nodes and 5 edges was found by Larry Moss . Another example with the same number of nodes and edges was found independently by Scott Johnson. It is the following graph:

4.28 Corollary:

A FA , FA FA and SAFA are pairwise incompatible axioms.

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I Another Variant

5

Boffa ' s Weak Axiom

Recall that V z a is the subgraph of the system V having all the nodes in the set a and all the edges of V between those nodes. Note that V z a need not be extensional but certainly is when a is a transitive set ; Le. when x E a always implies that x � a . Let BAt be the following axiom : •

Every extensional graph is isomorphic to V z a for some tran­ sitive set a .

Notice the similarity between this axiom and Gordeev's axiom GA mentioned at the end of chapter 2 . The statement of GA involves a weakening of both the hypothesis and the conclusion of the above statement of BA l . In fact BA I is a good deal stronger than GA . We saw in chapter 2 that AFAI ==> GA. Here we will see in 5 . 3 that BAI is strictly stronger than AFAI . Call a set x a REFLEXIVE set if x = {x} . As any two reflexive sets are isomorphic it follows from FA FA2 , and hence from any A FA'" , that there is at most one reflexive set . This is in sharp contrast to the situation when BA I is assumed . For example, by considering the two-element extensional graph

0 0 we obtain a two-element set of reflexive sets . A set of reflexive sets of any cardinality can be obtained equally easily, so that we have: 5 . 1 Proposition: Assuming

BAI the reflexive sets form 57

a

proper class.

58

Variants of t he Ant i- Foundation Axiom

BA I may be viewed

as

giving the most generous possible an­

swer to the question Which apgs are exact pictures ? 5 . 2 Proposition: BA I is equivalent to:

an apg is

an

exac t picture iff it is extensional.

Proof: Note that every exact picture must be extensional by the axiom of extensionality. Let Ga be an extensional apg. Then by BAI there must be a transitive set c and b E c such that Ga � ( Vz c)b � Vb. Hence Ga is an exact picture. Conversely suppose that every extensional apg is an exact picture and let G be an extensional graph. There are two cases. First suppose that ac = G for some a E G. Then Ga is an extensional apg containing all the nodes of G . As Ga is extensional it is an exact picture so that Ga � Vc for some set c. As ac = G it follows that c must be a transitive set and that c E C so that G � Vzc. Second suppose that ac 1= G for all a E G . Then we can form an extensional apg G' * by adding a new node * to G and new edges ( * , a) for each a E G. As G' * is extensional it is an exact picture so that G' * � Vc for some set c. As a E *c' implies that ac � *C' it follows that c must be a transitive set and G � V z c. D

5 . 3 Corollary:

BAI

==>

A FAr & ..., A FA2' for any regular bisimulation ""'.

5.4 Exercise : Show that every graph is isomorphic to a sub­ graph of an extensional graph.

Recall from chapter 4 that Mo ::::; M if there is an injective system map Mo � M. Note that G ::::; V iff G � ( Vzc) for some transitive set c. Call an extensional system M a L O C A L LY U N IVERS AL system if G :j M for every extensional graph G. Observe that BA I is equivalent to: •

V

is locally universal.

5 . 5 Exercise: Show that is locally universal.

a

full system is

a

m od el of BAI

iff

it

Another Variant

59

Boffa ' s Axiom and S up eruniversal Syst ems

Here we will consider an axiom for non-well-founded sets due to M. Boffa. Assuming that V � On we will show that this axiom has a unique full model up to isomorphism. In this respect it is like the axioms A FA"' , but it turns out not to be one of these. The following notion will be useful when we come to formulate Boffa's strengthening of BA l . The system M is a TRANSITIVE SUBSYSTEM of the system M', abreviated M � M' , if M � M' and x M' = X M for all x E M. 5.6 Exercise: Show that

(i) M � M' iff M

� M' and the inclusion map M 7rz X "' M 7rz y - X "'G y. For x, y E G let xRy - 1/Jx 1/Jy. We must show that xRy - x "'G y. If xR + y then =

{1/Jy' I y' E Y G } {1/Jx' I x' E x G } 1/Jy, as G' is so that (1/Jx)G' (1/J Y )G' and hence 1/Jx + extensional. So R � R. As R is reflexive it follows that x "'G y ==} xRy. For the converse implication let xRy; i.e. 1/J x = 1/Jy. If 1/J x E G� then x , y E Go and 1/Jox 1/JO Y so that 7rX = try and hence x "' M y. It follows that x "' Go Y and hence x "'G y . Here we have made two further applications of part (iv) of exercise 5. 15. If 1/J x E G' - G� then x y so that , as "' G is 0 reflexive, x "'G y. G' is surj ective By this lemma and the fact that 1/J : G M' such that for x E G there is a unique inj ective map 7r ' : G' 7r( 7ri X ) . 7r ' (1/Jx) =

=

=

=

=

---+

---+

=

It is easy to check that 7r is a system map that extends the identity map on G�. 0

This page intentionally no longer blank

Part Three O n Using t he Anti- Foundat ion Axiom

This page intentionally no longer blank

6

I

F ixed Point s of

Set Continuous Operators

In this chapter we consider the notion of a set continuous opera­ tor. Each such operator will be shown to have both a least and a greatest fixed point . Set Continuous O p erators



6 . 1 Definition: Let be a class operator; i.e. X is a class for each class X. is SET CONTINUOUS if for each cl ass X

U{x I

X This

tions.

(1)

is equivalent to the conjunction

is



&

of

x

�

X}.

the following two condi­

MONOTONE; i. e. X�y

(2)

V

x E

==}

X � y.

is SET BASED ; i.e. a E X

==}

aE

x

for some set

x

� X.

Set continuity has alternative characterizations given following exercise. 6.2

Exercise: Let are equivalent. ing

be a class operator.

by

Show that the follow­

( i ) is set continuous. (ii) There is a class relation R such that for all classes X X

=

{a I 3x E V aRx 73

the

&

x � X}.

On Using the Anti-Foundation Axiom

74

( iii ) There is a map /I

: Ll

-+

(Th )h E A of maps Th : Vl/ h

4>X

n

=

V, for some class �, and a family V such that for all classes X

-+

{ Th f I 6 E � & f E X l/ t5 } .

Obvious examples of set continuous operators are for each class A, where for each class X

pow ,

Id

a d KA

pow X

=

Id X

KA X

{x E V I x � X } , X, A.

Also the composition 4> 0 \II of two set continuous operators 4>, \[J is clearly set continuous . For any system M we have the following example of a set continuous operator used in the construction of the maximal bisimulation on M. For each class X 4> M X is the class of pairs (a , b) E M x M such that 'ix

E aM

3y E bM (x, y)

E

X & 'iy E bM 3 x E

aM

(x, y) E X

The next exercise details some ways of constructing new tinuous operators out of old ones.

set

con­

6.3 Exercise: Let (4)d i E I be a family of set continuous opera­

tors indexed by the class I.

(i) Show that EiE I 4> i is set continuous, where for each class X iE I

( ii )

( iii )

iE !

If I

is a set show that D i E ! 4>i is set continuous, where for

If I

=

each

class

X

{ I , . n} show that 4>1 . .

x . . , x

4>n is set continuous,

where for each class X

Note that for set continuous 4> 1 , . . . 4>n we can set continuous operator 4>1 + . . . + 4> n , where

also

define the

Fixed Points of Set Continuous Operators

75

I}>l + . . . + I}>n = L iE I I}> i when f = { I , . . . , n} . Also if I}> is set continuous then so is I}> I for each set f, where I}>I = ITi E I I}>i when I}> i = I}> for each i E f. Using the results of this exercise a great variety of set contin­ uous operators can be formed. An example, chosen more or less at random, is the set continuous operator I}> = (pow ( (powfd) + fdI )) X KA , where f is a set and A is a class. This is the operator such that for each class X I}>X

for all classes

pow (pow X + Xl)

X

A

X.

F ixed Point s

We now turn to the construction of the least and greatest fixed points of a set continuous operator. If I}> is a set continuous { I i I ( I, -< , i) E B } where B is the class operator let fq, of triples (I, - { Jk I

k

-
is a set continuous operator and f = fq, then

( 1 ) I}>f

X f. Then, as I}> is set based, there is a set that a E I}>x and x is a set continuous operator let Jep

=

U {x

E

V I

x

� 4>x}.

6.5 Theorem: If IJ> is a set continuous operator and J

=

Jep

then

( 1) J � q,J , (2 ) If X � 4>X then X � J, (3) J is the largest fixed point of 4> .

Proo/"

( 1 ) Let

a E J . Then a E x for some set x such that x � 4>x. It follows that a E 4>J as x � J and IJ> is monotone. (2 ) Let X � 4>X and let a E X. We must show that a E J. We first show that for each set x � X there is a set x' � X such that x � IJ>x' . So let x � X. Then x � 4>X so that

Vy E

x

3u

y

E lJ>u & u � X.

By the collection axiom scheme there is a set Vy

E

x

3u

A

E A y E q,u & u � X.

such that

we let x ' = U{ u E A I u � X} then x ' is a subset of X and x � IJ>x' as required. Now we can use the axiom of dependent choice s to find of subsets of X such that an infinite sequence xo , X l , If

.

•

•

Fixed Points of Set Continuous Operators

77

Xo = {a} and Xn � cJ>Xn + 1 for all n. Let x Un Xn . Then x is a set and if y E x then y E Xn for some n so that y E Xn � cJ>Xn + 1 � cJ>x. Thus x � cJ>x. As a E Xo � x it follows that a E J. I do not know if this use of the axiom of =

dependent choices was essential. (3) The argument to show that J is the largest fixed point of cJ> is simply dual to the argument , at the end of the proof of the previous theorem, that I is the least fixed point . 0 In certain cases the fixed points Iq, and Jq, of a set continuous operator cJ> are equal. In these cases Iq, is the unique fixed point of cJ>. For example if we assume the axiom of foundation then V is the unique fixed point of pow and 0 is the unique fixed point of cJ> where cJ> X = A x X for all classes X . Of course when A FA is assumed pow and cJ> have many fixed points. Recall that Ipow = VwJ while Jpow = V . Also Iq, = 0 while Jq, is the class of all streams (ao , (aI , (a2 , . . . ) ) ) of elements ao , aI , a2 , . of A . The following gives a sufficient condition for a set continuous operator to have a unique fixed point . .

.

6 .6 Exercise: Let cJ> be a set continuous operator such that

there is a well-founded class relation

X and all a E cJ>X a E cJ>{x E X I x Show that Iq,

=

-
be set continuous. Working in ZFC- show that there are classes p� and JOt , for a E On, so that

78

On Using the Ant i- Foundat ion Axiom

Show also that Iq, Jq,

U

aEOn

r:t ,

n J aEOn

a

.

Often a set continuous operator has the following additional property. 6.8 Definition: The class operator cJ> PRESERVES INTERSEC­ TIONS if for every family of classes (XdiEI iEI

iE I

If the set continuous operator cJ> does preserve intersections then cJ> (nn<w In ) = nn <w cJ> J n = nn<w I n . It follows that this is the largest fixed point Jq, . 6.9 Exercise: Show that:

(i ) If cJ> is defined

as

in (ii) of exercise 6.2. and for all a, x, Y

a Rx & aRy

==}

x

=Y

then cJ> preserves intersections. (ii) If cJ> is defined as in (iii) of exercise 6.2. and for all 01 , 02 E and all II : VOl - V, h : V02 - V

�

then cJ> preserves intersections. We end this chapter with a useful application of AFA. We use the terminology of the Substitution and Solution lemmas of chapter 1. So let X be a class of atoms and let cJ> be a set continuous operator with largest fixed point J . We call an X­ set a a cJ>-LOCAL set if for every class B of pure sets and every T:X-B f'a E cJ>B.

Fixed Points of Set Continuous Operators

79

6 . 1 0 Theorem: (assuming AFA) Let ax be a and 'll is a standard functor. The following exercise gives further ways to construct new standard functors from old ones.

Let (4)i )i E I be a family of standard functors indexed by the class I. 7.4 Exercise:

(i) Show that 2:iE I 4>i is

class

and

a

standard functor 4> where if X is a

84

On Using the Anti-Foundation Axiom

if 1f : X

Y and (i , a) E cl>X.

-

II cl>i

(ii) Show that if I is a set then

is a standard functor cI>

iEI

where if X is a class

and ( (cI>1f) f)i

=

(cI> i1f) (Ji)

if 1f : X - Y, J E cl> X and i E I. (iii) Show that if I = { i , . . . , n } then cl>l functor cI> where if X is a class

x . . . X

cl>n is a standard

and ( cI>1f ) (a 1 , " " an )

if 1f : X

-

=

( (cI> l 1f)a l , . . . , (cI>n1f) an )

Y and ai E cl>iX for i

=

1 , . . . , n.

7.5 Exercise: Assume that the set continuous operator cI> is

defined in terms of a family ( 76 ) 6 E � 6.2; i.e. for each class X cI> X

in part (iii) of exercise

as

{ 76 f I 8 E A & f E Xv 6 } .

=

Assume that

762 (1f 0 h ) whenever 1f : X - Y, 81 , 82 E A and It E X V61 , h E X V 62 . 761 It

Show that

=

cI>

whenever 1f : X

762 h

===*

761 ( 1f It ) 0

=

can be made a standard functor by defining

-

Y, 8

E

Ll

and f

E

x V6 .

We shall need to use the following If 1f : X - Y and Z � X then

property of standard func­

tors.

cI>(1f r Z) = (cI> 1f) r ( cI>Z) . This

is

because

1f

rZ

=

1f

0

iz, x .

The S pecial Final Coalgebra Theorem

85

Also note that any fixed point of a standard functor can be viewed as a full algebra or full coalgebra, using the identity map. We have the following result concerning the least fixed point. 1.6 Theorem: If ip is a standard functor then 1$ is an initial algebra.

Proof: Let (A, a) be an algebra. We must show that there is a unique homomorphism from 1$ to ( A , a) ; i.e . , a map 7r : 1$ ----+ A such that for all x E 14> 7rX

=

a

( ( ip 7r) x ) .

Recall that 1$ = U). E On IA , where IA = ip( U d IlL ) . (See exercise IL 6.7) Note that IlL � IA for J.L < A. We will define 7rA : IA ----+ A, by transfinite recursion on A E On, so that for J.L < A 7rIL = 7rA r IlL . The problem of checking that this definition can be carried out in ZFC- will be ignored here. The desired map 7r will be defined as the union of the 7rA • As induction hypothesis we assume that 7r IL : IlL ----+ A has been defined for all J.L < A so that 7rIL r r for v < J.L < A . Let Id = U ILd IlL and 7r d = U d 7rIL . By our induction hy­ IL pothesis 7r such that for all iEA , , . 7rI Z· = 7rXi = 7rax, = 7r Ca i = ( 7r ' ) ( O: Z ) , so that the diagram

90

O n Using the Anti- Foundat ion Axiom

A

7r

'

J4>

II cl> 7r ' commutes. This means that 7r ' is a homomorphism from the coalgebra (A, a) to the co algebra Jip . As the solution 7r is unique, 0 it follows that the homomorphism 7r ' is unique. In practice the natural functors always seem to be uniform on maps. For example to see that pow is uniform on maps let bu = { 2 } X {X i l i E u} for u E pow I. It is also the case that the natural standard functors always seem to preserve weak pullbacks. Note that while pow does preserve weak pullbacks it does not preserve pullbacks. It would be interesting to sort out the relationship between the two notions "uniform on maps" and "preserving weak pullbacks" .

8

I An Application to Communicating Systems

In this chapter we illustrate some of the general theory described in the previous two chapters by considering the example from computer science of Robin Milner's Synchronous Calculus of Communicating Systems , abbreviated sees. (See Milner 1 983 .) This calculus can be viewed as a mathematically streamlined and synchronous version of the earlier calculus ees. (See Mil­ ner 1980. ) In (Milner 1 983) Milner set up sees by giving an inductive definition of a class of infinitary expressions. These ex­ pressions are intended to represent the possible states of systems that can communicate with each other. Communication between systems is represented by synchronisation of atomic actions. To capture the idea of synchronisation Milner uses an Abelian group Act of atomic actions. The parallel synchronous composition of two atomic actions a, b is represented by the atomic action ab obtained by using the group operation to compose a and b. The identity aa- I = 1 , where 1 is the unit of the group, is used to represent the synchronisation of an atomic action a in one system with the inverse atomic action a-I in another system. Here we take the view that this aspect of sees is not fundamental to its mathematics. So we will assume given an arbitrary set Act of atomic actions and impose no structure on it. O f course in the applications Act will need to be structured suitably, but such structure can be introduced as needed. Milner gives the expressions of sees an operational seman­ tics in terms of an inductive definition of a family of binary relations on the class of expressions. These relations are indexed by the set Act and used to represent allowed transition steps from one state of a system to another , each step being labelled with an element of Act. So the operational semantics determines what 91

92

On Using the Anti-Foundation Axiom

has been called a labelled transition system. Different expressions of sees can have the same abstract behaviour as determined by the operational semantics. In order to capture this notion of abstract behaviour Milner makes use of a concept first consid­ ered by David Park in ( Park 198 1 ) . This is the concept of a bisimulation relation on a labelled transition system. Among the bisimulation relations there is always a maximal one, which is moreover an equivalence relation. The notion of bisimulation re­ lation on a system used in this book is simply the special case of Park's notion when Act is a singleton set. Milner calls the maximal bisimulation relation on the expressions of sees strong congruence. The final step of Milner's construction is to form a quotient of the class of expressions by strong congruence. The re­ sult is a labelled transition system which gives a model of abstract behaviours for a certain notion of computational system. As we will see, transition systems labelled by elements of a set Act can be viewed as coalgebras relative to the standard functor pow(Act x . . . ) and Milner's quotient construction then becomes a construction of a final co algebra relative to that functor. In fact Milner's quotient construction was the prototype for a proof of the final coalgebra theorem. As final coalgebras are unique up to isomorphism when they exist, only the existence of a final coalgebra is of any purely mathematical concern. In fact the final co algebra theorem applies to the functor, as it preserves weak pullbacks. When Act is a singleton set then the functor is isomorphic to the functor pow whose final coalgebras are the complete systems used to model AFA . It was the initial perception of this connec­ tion between Milner's construction and set theory that has led to the author's interest in non-well-founded sets and the work presented in this book. As j ust described , the connection between sees and A FA is that the model construction for A FA is simply a special case of the quotient construction for sees. Put another way, sets in the A FA-universe are the abstract behaviours for the special case of sees where there is only one atomic action. In fact it is more profitable to reverse the connection between sees and A FA by making use of the special final coalgebra the­ orem. That theorem applies to the functor pow (Act x . . . ) , as this standard functor is easily checked to be uniform on maps.

An Application to Communicating Systems

93

It follows that the largest fixed point of the functor is a final coalgebra, provided that we assume AFA . In this way we get a very simple and direct set theoretical construction which can be used to replace Milner's considerably more elaborate quotient construction. Of course the "penalty" to be paid for this simplic­ ity is the need to use non-well-founded sets and A FA. But if one accepts the point of view suggested in this book then that is no penalty at all. Transit ion Systems

Transition systems form a natural model for computation pro­ cesses. Such systems consist of a class X of possible states of the system and a family of binary transition relations � between states, one for each possible atomic action a of a process. So x � y holds if there is a possible atomic transition step of the process from the state x to the state y in which the atomic action a takes place. So a computation of a process starting in a state Xo will have the form X o � Xl � X 2 � . . . where x o , X l , X 2 , . . . are the successive states that the process passes through and a o , a I , a 2 , . . . are the atomic actions performed at successive steps of the computation. These atomic actions are intended to rep­ resent what is externally observable, while the successive states that a process passes through in a computation are intended to be internal to the process. So distinct states of processes may have the same external behaviour . Transition systems may be conveniently represented as pairs (X, a) where X is a class and a ; X ---+ pow(Act x X) is the map given by;

ax = { (a, y ) E A ct

x

XI

X

�

y}

for al l X E X. The transition relations can b e recaptured from a using the definitions: X

--+ Y a

�

(a, y) E ax

for x, y E X. Now observe that a : X --+ ex where following standard functor on the category of classes: e

=

pow

0

(KA ct

x

e

is the

Jd ) .

So from now o n we will take a TS; i.e. a TRANSITION SYSTEM , to be a coalgebra relative to this functor e , with transition relations

94

On Using t he Anti-Foundation Axiom

as determined above. Notice that we allow a TS to have a class of states and so will call it small if the class is in fact a set. We will call a coalgebra homomorphism between TSs a TS map. The C omplete Transition System

P

As 8 is a standard functor that preserves weak pullbacks we may apply the final coalgebra theorem to get the existence of a final coalgebra for 8. We will call such a co algebra a C O M­ PLETE TS. The abstract behaviours of sees turn out to be t he states of a complete TS, a mathematical structure that is uniquely determined up to isomorphism. So sees could be de­ veloped axiomatically on the basis of a postulated complete TS. Here we prefer to follow an alternative course and instead use A FA and the special final coalgebra theorem. As the functor 8 is uniform on maps we can apply the theorem to get a simple set theoretical definition of a complete TS P. P is defined to be the largest fixed point of 8, or equivalently, it is the largest class such that if P E P then P is a subset of Act x P. P is a TS with transition relations � for a E Act given by P�Q

¢=}

(a, Q ) E P

for all P, Q E P. As P is a complete TS, for each TS (X, a ) there is a unique TS map (X, a ) ----t P. We will call this map the BEHAVIO U R M A P for (X, a) . It is the unique map 1f : X ----t P such that for

all x E X

1fX

=

{( a , 1fY ) I x � y in (X, a ) } .

If a TS arises as an operational semantics for a programming language then the behaviour map for the TS will give a canoni­ cal representation of the abstract behaviours of the programs of the language, as given by the operational semantics. In this way the complete TS P plays the role of a domain of mathemati­ cal objects that can be the denotations of programs for such a programming language. Some Operations on

P

We will define some operations on P that correspond to the four fundamental combinators that Milner used in ( Milner 1983) to define the expressions of sees. The four combinators were called Action, Summation, Restriction and Product .

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95

Action We start with the action operations. Given a E Act there is an operation on P that assigns to each P E P a set a : P E P such that for all b E Act and Q E P b

[a=b & P

a : P ---+ Q

=

Q J.

So a : P allows only the atomic action a to become P . In fact we define a : P = { (a, P) } . Summation Next we consider the summation operations . Given P� E P for i E I, where I is a set , there is a unique element LIE! Pz of P such that for all b E Act and Q E P

[ Pi � Q for some i E I ] . In particular, when I = 0 we get the null element 0 of P which allows no atomic steps, and when I = { I , . . . , n} we get the finite sum PI + . . . + Pn . In fact in general we define

and in particular we get that o

0

PI + . . . + Pn = PI U . . . U Pn . Note that the following equations trivially follow from these def­ initions. P+Q=Q+P P + ( Q + R) = (P + Q) + R P+O=P P+P= P There are also equations for indexed sums that we do not bother to state as they are simply the expected equations for indexed unions of sets. See (Milner 1983) where such equations play an important role in understanding sees as a calculus.

96

On Using the Anti- Foundation Axiom

Restriction

The third operation on P that we define is the restriction opera­ tion. Given A � Act there is a unique operation r A : P ---+ P such that for all P E P if b E Act and Q E P then P r A � Q if and only if b E A and P � pI for some pI E P such that pI r A = Q . To see this we consider the TS (P, Q A ) where Q A : P ---+ pow(Act x P) is given by -

QA P

=

P n (A

for all P E P . Then we can define unique behaviour map for (P, QA ) .

-

x

r

P)

A

:

P

---+

P to be the

Product

The product operation x is dependent on a binary composition operation assigning ab E Act to a, b E Act . It is the unique binary operation on P such that for H , P2 E P if b E Act and Q E P then H x P2 � Q if and only if

for some a I , a2 E Act such that a l a2 = b and some Q l , Q 2 E P such that Q I x Q2 = Q . In fact we can define x to be the unique behaviour map for the TS (P x P , , ) where , : P x P ---+ pow (Act x (P x P)) is given by ,( H , P2 )

=

{ (a l a2 , ( Q I , Q 2 ) ) I H � Q l & P2 � Q2 }

for all H , P2 E P . This completes the description of the operations on P corre­ sponding to the four fundamental sees combinators in (Milner 1 983) . Milner also considers two further derived combinators, morphism and delay. We give the operations on P that corre­ spond t o them . Morphism

For each map 'P : Act ---+ Act there is a unique morphism op­ eration - ['Pl : P ---+ P such that for all P E P if b E Act and

Q E P then P['Pl � Q if and only if P � p' for some a E Act such that 'Pa = b and some pI E P such that P' ['Pl = Q. In

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97

fact it is the unique behaviour map for the TS (P, (3